Re: Bayer's Theorem

The idea is that a variable might be "correlated" with another. I.e. they are not INDEPENDENT. For example, the tallness and body weight of high school students (it is obvious that the taller one is, the more they are likely to weight).

Assume the probability of weighting between 65-70 kg is 50% among our students.The probability of weighting between 65-70 kg - given you are more than 185 cm tall - will not presumably be 50% any more: most of the students taller than 185 cm will also weight more than 70 kg.

So, assume the probability of weighting between 65-70 kg, given you are >185 cm tall, is 30%.What is then the probability of both weighting 65-70 kg and being >185 cm tall?

Assume 10% of the students are >185cm. We said that 30% of them will weight between 65-70 kg.Then the probability of both events will be: 10% x 30%, 0.1x0.3 = 0.03 = 3%.Notice that this differs from the simple product of the two probabilities (50% probability of weighting between 65-70 kg) x (10% probability of being >185 cm tall) = 0.05 = 5%. This would be the probability of both events IF weight and tallness were INDEPENDENT. Bu they are not.

But, quite obviously, the probability of W AND H also = probability of H given W.

This, may seem trivial and useless, but it is not so, absolutely.

Get a blood test for a disease. You turn out positive.What is the probability that you actually have the disease?In order to tell it, you will have to know that the disease has a prevalence of 0.1% (it affects 1:1000 of the population), and that the test is positive in 90% of the affected subjects, whereas it is positive in 10% of the unaffected people (false positives).Overall, you can deduce that the probability of getting a positive result, p(+), equals 90% x 0.1% + 10% x 99%(i.e. 90% of the diseased people + 10% of the unaffected people) = 0.0009+0.099 = 0.0999 = 10%.Out of 1000 people, 0.9 will be diseased and have a positive test; 99 will have a positive test but will not be affected.

Re: Bayes Theorem

It is extraordinary that people like Thomas Bayes had the capacity to envision, experiment and deduce these useful relationships.

But it is also just as extraordinary that someone might go through Bayes' papers and recognize something of value. There are actually two stories here. One of discovery (Bayes), and one of the discovery of the discovery (Bayes' Friends). What are the chances of that (asked the Elemental)?